Nikulin determined [1]} which abelian groups \(G\)  can appear as automorphism groups of some complex K3 surface \(X\)  that act trivially on \(H^{2,0}(X)\) .
In our setting these are precisely the abelian subgroups of \(\operatorname{Diff}(M)\)  that fix a positive 3-plane in \(\mathbf {\Lambda }_\mathbb {R}\)  and some complex structure on \(M\) .
If we assume that \(G\)  is cyclic of prime order \(p\) , then the primes \(p\)  that occur are \(2,3,5,7\) . A more detailed study for the case \(p=2\)  was given by Morrison [2]} and Van Geemen-Sarti [3]} and for the odd primes by Garbagnati-Sarti [4]}. There is in fact a very helpful picture in terms of an elliptic fibration that enables us to get a good understanding of the topology of such an action. We will say more about this in Section REF .
